1. Data and modeling issues in population biology  Alan Hastings (UC Davis) and many colalborators  Acknowledge support from NSF

2. Goals  Understand ecological principles or determine which processes are operating –E.g., How important is competition?  Make predictions –What will the population of a species be in the future?  Management –Fisheries –Infectious diseases –Invasive species –Endangered species

3. Time series  Population biology typically follows populations through time (and sometimes space)  Data is of varying qualities Limited extent Measurement error Often cannot go back and get more data

4. Time series of total weekly measles notifications for 60 towns and cities in England and Wales, for the period 1944 to 1994; the vertical blue line represents the onset of mass vaccination around 1968. (Levin, Grenfell, Hastings, Perelson, Science 1997)

5. A small part of the Coachella valley food web (Polis, 1991)

6. Purposes of time series analysis  Parameter estimation –For a ‘known model’, estimate the parameters –Determine importance of biological factors operating  Model identification –The? model with the ‘highest likelihood’ is chosen as the model that describes the system  Prediction  Management

7. Underlying modeling issue  Mechanistic models versus using general models –Linear time series analysis  Less of an issue –Nonlinear time series  Use general model  Use specific model  Use ‘mixed’ model

8. Modelling approaches  General model –E.g. cubic splines  Mechanistic model –E.g., SIR model  Mixed –TSIR  Know that a single infection removes a single susceptible, and know dynamics of I to R whereas S to I is more problematic

9. How ‘noise’ enters  Process noise –Environmental variability –Role of species or factors not included  Demographic factors –How mechanistic should this be?  Other species, or environment  Measurement error –How good are population estimates?  How mechanistic should this be?

10. The time series True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Dynamics + ‘noise’ Observation process, possibly with error

11. Use Kalman Filter True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Linear Dynamics + ‘noise’ Linear Observation process, possibly with error

12. Observation error only True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Dynamics + ‘noise’ Observation process, possibly with error Use model to generate the whole time series, minimize difference between every observation and every prediction

13. Process error only True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Dynamics + ‘noise’ Observation process, possibly with error

14. Process error only True population at time t Observed = true population at time t + 1 Predicted population at time t + 1 Dynamics + ‘noise’ Dynamics only Noise Make one step ahead predictions only Minimize difference between one step ahead and observation

15. Laboratory populations of Tribolium

16. Graphical representation of Dungeness crab model

17. Data and fits at eight ports

20. Resample from ‘noise’ to demonstrate that observed dynamics result – essentially continual transients

28. Conclusions  Much more work needed  Mechanistic models can be used  Using mechanistic models can be important in highlighting ecological processes

29. Definition of state space model ‘true’ population dynamicsnoise Observed population Observation process noise

30. The time series True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Dynamics + ‘noise’ Observation process, possibly with error

31. Likelihood is defined as probability of observation LikelihoodProbability defined iteratively Superscripts on y’s mean observations up to and including that time, subscripts denote observations only at that time parameters

32. Begin iterative calculation of likelihood  Probability of first observation is found by summing the probabilities of all possible first states times probability of observation given state  Then adjust distribution of states to reflect first observation

33. The time series True population at time t True population at time t + 1 Observed population at time t Observed population at time t + 1 Dynamics + ‘noise’ Observation process, possibly with error

34. Second and remaining steps

35. Later steps

36. Change to continuous state

37. Replace sums by integrals and pdf’s

38. Compute first state as stationary distribution

39. Change all computations to computations of pdf’s  Omit details

40. Use Beverton-Holt and Ricker models with process noise linear on log scale

41. Assume observation noise is linear Noise structure could be more general

42. Dynamics and fitting  Beverton-Holt –Always stable –Use one set of parameter values  Ricker –Period doubles, etc –Stable equilibrium –Two cycle –Four cycle

43. Process noise and observation error combinations  Large process noise, small observation error  Small process noise, large observation error  Large process noise, large observation error  Generate 300 time series of length 20 for each of the 12 cases (3 error structures by 4 model structures)

44. Parameter estimation  For each case, use each method  NISS should handle large noise  LSPN (least squares process noise)  LSOE (least squares observation error)

45. Maximum likelihood estimates of growth rate parameters

47. Estimate of growth rate when data generated by one model, fit by another Top row, generated by BH, fit by Ricker, bottom row is reverse

48. Now, model identification  Generate data with either model, see which model has the highest likelihood

49. ‘true’ model incorrect correct incorrect

52. Conclusions  NISS is much better at parameter estimation –Note computational intensity  Model identification problem is hard –Tendency to pick ‘more flexible’ model –Made easier with more noise or more complex dynamics –Does it matter when dynamics are simple? –Difficulty in ecology of predicting ‘out of sample’

56. References  Higgins, K., A. Hastings, J. N. Sarvela, and L. W. Botsford. 1997. Stochastic dynamics and deterministic skeletons:population behavior of Dungeness crab. Science 276:1431–1435.  de Valpine, P. and A. Hastings. 2002. Fitting population models incorporating noise and observation error. Ecological Monographs 72:57- 76.